Chaotic orbits for spinning particles in Schwarzschild spacetime

We consider the orbits of particles with spin in the Schwarzschild spacetime. Using the Papapetrou-Dixon equations of motion for spinning particles, we solve for the orbits and focus on those that exhibit chaos using both Poincaré maps and Lyapunov exponents. In particular, we develop a method for comparing the Lyapunov exponents of chaotic orbits. We find chaotic orbits for smaller spin values than previously thought and find chaotic orbits with astrophysically relevant spin values.

Figure 1

Three Poincaré sections of the phase space trajectories of nonchaotic particle orbits. Each has $L=3.6$ and from the innermost orbit out $E_1=0.9499$, $E_2=0.9511$, and $E_3=0.9534$. Notice that the trajectories are confined to the surfaces of the tori which intersect the section.

Figure 2

A close-up view of the orbits from Fig. 1. Notice that this zoomed in view continues to show the trajectories are constrained to the tori.

Figure 3

Three Poincaré sections of the phase space trajectories of chaotic particle orbits. From the inside orbit out the orbits have spin values $S_1=1.515$, $S_2=1.581$, and $S_3=1.627$. Notice that the trajectories stay close to the surface of the tori similar to Fig. 1, but do not have the thinly defined surface.

Figure 4

A close-up view of the orbits from Fig. 3. Notice that this zoomed in view continues to show the trajectories are not constrained to the tori. Compare to Fig. 2.

Figure 5

We see the jagged Lyapunov function converging to zero for an orbit with $S=0$. We also plot the curve fit to the data. This orbit begins at $r=6m$ with $P_{\varphi }=3.6$. The predicted Lyapunov exponent for this orbit is $-1.68\times {10}^{-5}$ with an rms error of $1.23\times {10}^{-5}$.

Figure 6

We see the jagged Lyapunov function converging to a nonzero value for an orbit with $S=0.5$. The initial conditions are $r=6m$, $P_{\varphi }=3.6$, $\alpha =\frac{\pi }{4}$, and $\beta =\frac{\pi }{4}$. We also plot the curve fit to the data. The predicted Lyapunov exponent is $3.787\times {10}^{-4}$ with 0 well outside the rms error, $6.1\times {10}^{-5}$, of the fit.

Figure 7

These Lyapunov exponents are given by the constant term in the curve fitting model, Eq. (28), with root mean square error bars. The initial spin orientation is $\alpha =\frac{\pi }{4}$ and $\beta =\frac{\pi }{4}$ and constants of the motion for the spinless case are $L=3.6$ and $E=0.9522$. Compare with Figs. 8, 10. Note that the orbits transition from nonchaotic to chaotic around a spin of $S=0.5$.

Figure 8

These Lyapunov exponents are given by the constant value in the curve fitting model with the root mean square value providing the error bars. The initial spin orientation is $\beta =\pi$ and constants of the motion for the spinless case are $L=3.6$ and $E=0.9522$. All spin values higher than $S=0.3$ cause the particle to cross the event horizon. Note that lower spin values give exponents of the same order of magnitude as those in Fig. 7.

Figure 9

These Lyapunov exponents correspond to the small spin case of Fig. 8 with $\beta =\pi$. Note the monotonically decreasing, yet still positive, values of the exponent for decreasing spin values. This trend continues for spins as low as $S=0.015$. However, this trend would appear to be lost for the very smallest values of spin. Indeed, we feel confident to treat these values as zero with correspondingly nonchaotic orbits.

Figure 10

These Lyapunov exponents are given by the constant value in the curve fitting model with the root mean square value providing the error bars. The initial spin orientation is $\beta =0$ and constants of the motion for the spinless case are $L=3.6$ and $E=0.9522$. No large exponents (compare to Fig. 7) appear in this configuration.

Figure 11

These Lyapunov exponents are given by the constant value in the curve fitting model with the root mean square providing the error bars. The initial spin orientation is $\beta =0$ and constants of the motion for the spinless case are $L=3.6$ and $E=0.9522$. Note that unlike Fig. 7 the exponents all appear to be nonzero. No large exponents (compare to Fig. 7) appear in this configuration. Note the continuous nature to the exponents and the apparently positive value for spin as low as $S=0.01$.

Figure 12

These Lyapunov exponents correspond to knife edge orbits. Notice that the $S=0$ case is very nonzero. This is referred to as a “chaos mimic.” The spinless orbit begins at $r=100m$ and has three inner loops at small radius for each large scale orbit. The initial spin orientation of $\beta =0$. In the spinless case $E=0.997$ and $L=3.9246$.

Figure 13

These Lyapunov exponents correspond to the same knife edge orbits as Fig. 12. Notice that zero is outside the error bars for spins as low as $S=0.03$.

Figure 14

This is a plot of the effective potential corresponding to a knife edge orbit constrained to high curvature regions. This orbit has $L=3.47$ and $E=0.9432$. The line $E^2=0.8897$ is shown by the dotted line and shows that the particle is constrained to remain near $r=6m$.

Figure 15

Lyapunov exponents for spin values in the physical range. These orbits have orientation $\beta =\pi$ and constants of the motion for the spinless case are $L=3.47$ and $E=0.9432$. Note the chaos mimic when $S=0$ and the magnitude of the exponent is comparable to the larger values of Fig. 7.

Figure 16

These are Poincaré sections in the $r-P_r$ plane for $S=0$ and five orbits with spin magnitudes spaced equally from $S=3.0\times {10}^{-5}$ to $S=3.069\times {10}^{-5}$. These orbits have the orientation $\beta =\pi$ and constants of the motion for the spinless case are $L=3.47$ and $E=0.9432$. Notice that unlike Figs. 1, 3 the initial conditions of the orbits make them difficult to distinguish at this level.

Figure 17

This is a closer look at Fig. 16. The point $r=5.67$ is marked on the axis which spans $r=5.67-1\times {10}^{-6}$ to $r=5.67+3.5\times {10}^{-6}$. This finer line in the upper left is the part of the section for the spinless case. Notice the distinct breaking of the tori for the other five orbits. These orbits increase in spin from top to bottom of the upper left corner.